Question: Solve for $x$ : $ 7|x - 4| - 6 = 5|x - 4| + 4 $
Explanation: Subtract $ {5|x - 4|} $ from both sides: $ \begin{eqnarray} 7|x - 4| - 6 &=& 5|x - 4| + 4 \\ \\ { - 5|x - 4|} && { - 5|x - 4|} \\ \\ 2|x - 4| - 6 &=& 4 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 2|x - 4| - 6 &=& 4 \\ \\ { + 6} &=& { + 6} \\ \\ 2|x - 4| &=& 10 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x - 4|} {{2}} = \dfrac{10} {{2}} $ Simplify: $ |x - 4| = 5$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 4 = -5 $ or $ x - 4 = 5 $ Solve for the solution where $x - 4$ is negative: $ x - 4 = -5 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& -5 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& -5 + 4 \end{eqnarray} $ $ x = -1 $ Then calculate the solution where $x - 4$ is positive: $ x - 4 = 5 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& 5 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& 5 + 4 \end{eqnarray} $ $ x = 9 $ Thus, the correct answer is $x = -1 $ or $x = 9 $.